C3 chapter 7 - further trigonometric identities and applications

In this chapter you will learn:
the angle addition formulae,
the double angle formulae,
the Rsin(θ+α) form and the angle factor formulae;
you'll practise proving identities using these and apply them to evaluating expressions and solving formulae.

Note:

Fluency in this chapter is essential for you to be able to solve a large proportion of the integration problems in C4 chapter 7.

C3 § 7.1

L.O. WALT recall and apply the addition formulae:

$\sin{(A+B)} \equiv \sin{A}\cos{B} + \cos{A}\sin{B}$

$\sin{(A-B)} \equiv \sin{A}\cos{B} - \cos{A}\sin{B}$

$\cos{(A+B)} \equiv \cos{A}\cos{B} - \sin{A}\sin{B}$

$\cos{(A-B)} \equiv \cos{A}\cos{B} + \sin{A}\sin{B}$

$\displaystyle \tan{(A+B)} \equiv \dfrac{\tan{A}+\tan{B}} {1-\tan{A}\tan{B}}$

$\displaystyle \tan{(A-B)} \equiv \dfrac{\tan{A}-\tan{B}} {1+\tan{A}\tan{B}}$

Examples:

C3 Exercise 7A

• C3 Exercise 7A worked solutions
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  C3 Exercise 7A (trigonometric addition formulae).jnt
  

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C3 § 7.2

Prior knowledge: it is expected that you have memorised the angle addition formulae by now.
WALT: derive and use the double angle formulae:

$\sin{(A+A)} \equiv \sin{A}\cos{A} + \cos{A}\sin{A} \implies \sin{(2A)} \equiv 2\sin{A}\cos{A}$

$\cos{(A+A)} \equiv \cos{A}\cos{A} - \sin{A}\sin{A} \implies \cos{(2A)} \equiv \cos^{2}{A} - \sin^{2}{A} \implies \cos{(2A)} \equiv 2\cos^{2}{A} - 1 \implies \cos{(2A)} \equiv 1 - 2\sin^{2}{A}$

$\displaystyle \tan{(A+A)} \equiv \dfrac{\tan{A}+\tan{A}} {1-\tan{A}\tan{A}} \implies \tan{(2A)} \equiv \dfrac{2\tan{A}} {1-\tan^{2}{A}}$

C3 Exercise 7B

• C3 Exercise 7B worked solutions
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  C3 Exercise 7B (double angle formula).jnt
  

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C3 § 7.3

C3 Exercise 7C

• C3 Exercise 7Cworked solutions
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  C3 Exercise 7C (using double angle trig in proofs).jnt
  

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C3 § 7.4

WALT convert expressions in terms of the addition of sines and cosines into the sines or cosines of a phase-shifted variable:
$a \cos{\theta} + b \sin{\theta} \equiv R \sin{\theta + \alpha}$

C3 Exercise 7D

• C3 Exercise 7D Q13 worked solutions
  

  C3 Exercise 7D (using Rsin(x + a) form to solve problems).jnt
  

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• C3 Exercise 7D Q14 visualisation:
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